We recall Gallai's decomposition theorem: if \(R\)  is finite, then \(R\)  is the lexicographic sum of the restrictions of \(R\)  to its components (hence, the strong modules of the index set of this sum are trivial). More importantly, if \(R\)  is a poset or a graph, all strong modules of \(R\)  are trivial if and only if \(R\)  is either prime or is a chain, a complete graph or has no edges ([1]} for the finite case, [2]} for the infinite case).
